Propensity rules in photoelectron circular dichroism in chiral molecules. II. General picture

Photoelectron circular dichroism results from one-photon ionization of chiral molecules by circularly polarized light and manifests itself in forward-backward asymmetry of electron emission in the direction orthogonal to the light polarization plane. To expose the physical mechanism responsible for asymmetric electron ejection, we first establish a rigorous relation between the responses of unaligned and partially or perfectly aligned molecules. Next, we identify a propensity field, which is responsible for the chiral response in the electric-dipole approximation, i.e., a chiral response without magnetic interactions. We find that this propensity field, up to notations, is equivalent to the Berry curvature in a two-band solid. The propensity field directly encodes optical propensity rules, extending our conclusions regarding the role of propensity rules in defining the sign of forward-backward asymmetry from the specific case of chiral hydrogen [Ordonez and Smirnova, Phys. Rev. A 99, 043416 (2019)] to generic chiral systems. Optical propensity rules underlie the chiral response in photoelectron circular dichroism. The enantiosensitive flux of the propensity field through the sphere in momentum space determines the forward-backward asymmetry in unaligned molecules and suggests a geometrical origin of the chiral response. This flux has opposite sign for opposite enantiomers and vanishes for achiral molecules.

Figure 1

Scheme of the right-hand side of Eq. (16) depicting the six different field geometries (circular blue arrows) in the molecular frame contributing to the net photoelectron current in the laboratory frame. For each field geometry only the component of the photoelectron current perpendicular to the polarization plane is taken into account.

Figure 2

Scheme of the right-hand side of Eq. (17) depicting the six different orientations of the molecular frame contributing to the net photoelectron current in the laboratory frame. The curved blue arrows indicate the field in the laboratory frame. For each orientation only the component of the photoelectron current perpendicular to the polarization plane is taken into account. These orientations are unique only up to a rotation around the axis perpendicular to the polarization plane.

Figure 3

Symmetry properties of an ensemble of chiral molecules interacting with circularly polarized light in the electric-dipole approximation. The ensemble is partially (or totally) aligned along the ($z$) axis perpendicular to the ($xy$) polarization plane of the light. The box represents the “enantiomer+field” system. Inside the box: the red letters $L$ and $R$ specify the enantiomer, the red double-headed vertical arrow specifies the direction along which the molecules are aligned, the blue curved arrow specifies the direction of rotation of a field circularly polarized in the $xy$ plane, and the golden vertical arrow stands for a polar vector observable $\vec{v}=v_z\widehat{z}$ displaying asymmetry with respect to the polarization plane $xy$. A reflection ${\widehat{\sigma }}_z$ with respect to the $xy$ plane leaves the field invariant but swaps the enantiomer and flips $\vec{v}$ (enantiosensitivity). A rotation ${\widehat{R}}_a^{\pi }$ by $\pi$ radians around any vector $\vec{a}$ contained in the $xy$ plane leaves the enantiomer invariant but swaps the polarization and flips $\vec{v}$ (dichroism). Note that a rotation ${\widehat{R}}_x^{\pi }$ (${\widehat{R}}_y^{\pi }$) followed by a reflection ${\widehat{\sigma }}_z$ is equivalent to a reflection ${\widehat{\sigma }}_y$ (${\widehat{\sigma }}_x$) and leaves $\vec{v}$ invariant but swaps both the enantiomer and the polarization. Thus, except for very specific cases (see Fig. 4), FBA in aligned ensembles is a signature of molecular chirality (see also Fig. 5).

Figure 4

Left: An achiral molecule consisting of four identical atoms with Cartesian coordinates $(-a,-b,b)$, $(a,b,b)$, $(a,-b,-b)$, and $(-a,b,-b)$. Right: Spatial inversion of the molecule on the left. A rotation by $\pi /2$ around the $x$ axis yields the molecule on the left. However, molecular alignment restricts available rotations. Thus if we consider a sample aligned along the vertical dotted line, the rotation by $\pi /2$ is not allowed and the aligned sample becomes effectively chiral.

Figure 5

Symmetry properties of an ensemble of chiral molecules interacting with circularly polarized light in the electric-dipole approximation. The ensemble is partially (or totally) aligned along an axis ($y$) contained in the ($xy$) polarization plane of the light. Notations are described in Fig. 3. This shows that, except for very specific cases (see Fig. 4), FBA in aligned ensembles is a signature of molecular chirality (see also Fig. 3).

Figure 6

Symmetry considerations for an ensemble of oriented (chiral or achiral) molecules. Notations are described in Fig. 3. The orientation axis ($x$) is in the polarization plane of the light ($xy$). A rotation ${\widehat{R}}_z^{\pi }$ by $\pi$ radians around the $z$ axis leaves the field invariant but flips both the molecular orientation and $v_y\widehat{y}$ (orientation sensitivity). A rotation ${\widehat{R}}_x^{\pi }$ by $\pi$ radians around the $x$ axis leaves the orientation invariant but swaps the polarization and flips $v_y\widehat{y}$ (dichroism). Note that a rotation ${\widehat{R}}_z^{\pi }$ (${\widehat{R}}_x^{\pi }$) followed by a rotation ${\widehat{R}}_x^{\pi }$ (${\widehat{R}}_z^{\pi }$) is equivalent to a rotation ${\widehat{R}}_y^{\pi }$ and leaves $v_y\widehat{y}$ invariant but flips the orientation and the polarization. For achiral molecules reflection symmetry forbids signals perpendicular to the polarization plane, i.e., $v_z=0$. For chiral molecules such FBA signal is not symmetry forbidden and it is enantiosensitive and dichroic.